Lecture 19: Chapter 8, Section 1 Sampling Distributions: Proportions
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1 Lecture 19: Chapter 8, Section 1 Sampling Distributions: Proportions Typical Inference Problem Definition of Sampling Distribution 3 Approaches to Understanding Sampling Dist. Applying Rule Cengage Learning Elementary Statistics: Looking at the Big Picture 1
2 Looking Back: Review 4 Stages of Statistics Data Production (discussed in Lectures 1-4) Displaying and Summarizing (Lectures 5-12) Probability Finding Probabilities (discussed in Lectures 13-14) Random Variables (discussed in Lectures 15-18) Sampling Distributions Proportions Means Statistical Inference Elementary Statistics: Looking at the Big Picture L19.2
3 Typical Inference Problem If sample of 100 students has 0.13 left-handed, can you believe population proportion is 0.10? Solution Method: Assume (temporarily) that population proportion is 0.10, find probability of sample proportion as high as If it s too improbable, we won t believe population proportion is Elementary Statistics: Looking at the Big Picture L19.3
4 Key to Solving Inference Problems For a given population proportion p and sample size n, need to find probability of sample proportion in a certain range: Need to know sampling distribution of. Note: can denote a single statistic or a random variable. Elementary Statistics: Looking at the Big Picture L19.4
5 Definition Sampling distribution of sample statistic tells probability distribution of values taken by the statistic in repeated random samples of a given size. Looking Back: We summarize a probability distribution by reporting its center, spread, shape. Elementary Statistics: Looking at the Big Picture L19.5
6 Behavior of Sample Proportion (Review) For random sample of size n from population with p in category of interest, sample proportion has mean p standard deviation shape approximately normal for large enough n Looking Back: Can find normal probabilities using Rule, etc. Elementary Statistics: Looking at the Big Picture L19.6
7 Rules of Thumb (Review) Population at least 10 times sample size n (formula for standard deviation of approximately correct even if sampled without replacement) np and n(1-p) both at least 10 (guarantees approximately normal) Elementary Statistics: Looking at the Big Picture L19.7
8 Understanding Dist. of Sample Proportion 3 Approaches: 1. Intuition 2. Hands-on Experimentation 3. Theoretical Results Looking Ahead: We ll find that our intuition is consistent with experimental results, and both are confirmed by mathematical theory. Elementary Statistics: Looking at the Big Picture L19.8
9 Example: Shape of Underlying Distribution (n=1) Background: Population proportion of blue M&M s is p=1/6=0.17. Question: How does the probability histogram for sample proportions appear for samples of size 1? Response: 5/6 1/6 0 1 Looking Ahead: The shape of the underlying distribution will play a role in the shape of for various sample sizes. for n=1 Elementary Statistics: Looking at the Big Picture L19.10
10 Example: Sample Proportion as Random Variable Background: Population proportion of blue M&Ms is Questions: Is the underlying variable categorical or quantitative? Consider the behavior of sample proportion for repeated random samples of a given size. What type of variable is sample proportion? What 3 aspects of the distribution of sample proportion should we report to summarize its behavior? Responses: Underlying variable Summarize with,, Elementary Statistics: Looking at the Big Picture L19.12
11 Example: Center, Spread of Sample Proportion Background: Population proportion of blue M&M s is p=1/6=0.17. Question: What can we say about center and spread of for repeated random samples of size n = 25 (a teaspoon)? Response: Center: Some s more than, others less; should balance out so mean of s is p =. Spread of s: s.d. depends on. For n=6, could easily get anywhere from to. For n=25, spread of will be than it is for n = 6. Elementary Statistics: Looking at the Big Picture L19.14
12 Example: Intuit Shape of Sample Proportion Background: Population proportion of blue M&M s is p=1/6=0.17. Question: What can we say about the shape of for repeated random samples of size n = 25 (a teaspoon)? Response: close to most common, far from in either direction increasingly less likely Elementary Statistics: Looking at the Big Picture L19.16
13 Example: Sample Proportion for Larger n Background: Population proportion of blue M&M s is p=1/6=0.17. Question: What can we say about center, spread, shape of for repeated random samples of size n = 75 (a Tablespoon)? Response: Center: mean of s should be p = (for any n). Spread of s: compared to n=25, spread for n=75 is Shape: s clumped near 0.17, taper at tails Looking Ahead: Sample size does not affect center but plays an important role in spread and shape of the distribution of sample proportion (also of sample mean). Elementary Statistics: Looking at the Big Picture L19.18
14 Understanding Sample Proportion 3 Approaches: 1. Intuition 2. Hands-on Experimentation 3. Theoretical Results Looking Ahead: We ll find that our intuition is consistent with experimental results, and both are confirmed by mathematical theory. Elementary Statistics: Looking at the Big Picture L19.19
15 Central Limit Theorem Approximate normality of sample statistic for repeated random samples of a large enough size is cornerstone of inference theory. Makes intuitive sense. Can be verified with experimentation. Proof requires higher-level mathematics; result called Central Limit Theorem. Elementary Statistics: Looking at the Big Picture L19.20
16 Center of Sample Proportion (Implications) For random sample of size n from population with p in category of interest, sample proportion has mean p is unbiased estimator of p (sample must be random) Elementary Statistics: Looking at the Big Picture L19.21
17 Spread of Sample Proportion (Implications) For random sample of size n from population with p in category of interest, sample proportion has mean p standard deviation has less spread for larger samples (population size must be at least 10n) n in denominator Elementary Statistics: Looking at the Big Picture L19.22
18 Shape of Sample Proportion (Implications) For random sample of size n from population with p in category of interest, sample proportion has mean p standard deviation shape approx. normal for large enough n can find probability that sample proportion takes value in given interval Elementary Statistics: Looking at the Big Picture L19.23
19 Example: Behavior of Sample Proportion Background: Population proportion of blue M&M s is p=0.17. Question: For repeated random samples of n=25, how does behave? Response: For n=25, has Center: mean Spread: standard deviation Shape: not really normal because Elementary Statistics: Looking at the Big Picture L19.25
20 Example: Sample Proportion for Larger n Background: Population proportion of blue M&M s is p=0.17. Question: For repeated random samples of n=75, how does behave? Response: For n=75, has Center: mean Spread: standard deviation Shape: approximately normal because Elementary Statistics: Looking at the Big Picture L19.27
21 Rule for Normal R.V. (Review) Sample at random from normal population; for sampled value X (a R.V.), probability is 68% that X is within 1 standard deviation of mean 95% that X is within 2 standard deviations of mean 99.7% that X is within 3 standard deviations of mean Elementary Statistics: Looking at the Big Picture L19.28
22 Rule for Sample Proportion For sample proportions taken at random from a large population with underlying p, probability is 68% that is within 1 of p 95% that is within 2 of p 99.7% that is within 3 of p Elementary Statistics: Looking at the Big Picture L19.29
23 Example: Sample Proportion for n=75, p=0.17 Background: Population proportion of blue M&Ms is p=0.17. For random samples of n=75, approx. normal with mean 0.17, s.d. Question: What does Rule tell us about behavior of? Response: The probability is approximately 0.68 that is within of : in (0.13, 0.21) 0.95 that is within of : in (0.08, 0.26) that is within of : in (0.04, 0.30) Looking Back: We don t use the Rule for n=25 because Elementary Statistics: Looking at the Big Picture L19.31
24 Rule (Review) For standard normal Z, the probability is 0.90 that Z takes a value in interval (-1.645, ) 0.95 that Z takes a value in interval (-1.960, ) 0.98 that Z takes a value in interval (-2.326, ) 0.99 that Z takes a value in interval (-2.576, ) Elementary Statistics: Looking at the Big Picture L19.32
25 Example: Sample Proportion for n=75, p=0.17 Background: Population proportion of blue M&Ms is p=0.17. For random samples of n=75, approx. normal with mean 0.17, s.d. Question: What does Rule tell about behavior of? Response: The probability is approximately 0.90 that is within (0.043) of 0.17: in (0.10,0.24) 0.95 that is within (0.043) of 0.17: in (0.09,0.25) 0.98 that is within (0.043) of 0.17: in (0.07,0.27) 0.99 that is within (0.043) of 0.17: in (0.06,0.28) Elementary Statistics: Looking at the Big Picture L19.34
26 Typical Inference Problem (Review) If sample of 100 students has 0.13 left-handed, can you believe population proportion is 0.10? Solution Method: Assume (temporarily) that population proportion is 0.10, find probability of sample proportion as high as If it s too improbable, we won t believe population proportion is Elementary Statistics: Looking at the Big Picture L19.35
27 Example: Testing Assumption About p Background: We asked, If sample of 100 students has 0.13 left-handed, can you believe population proportion is 0.10? Questions: What are the mean, standard deviation, and shape of? Is 0.13 improbably high under the circumstances? Can we believe p = 0.10? Response: For p=0.10 and n=100, has mean, s.d. ; shape approx. normal since. According to Rule, the probability is that would take a value of 0.13 (1 s.d. above mean) or more. Since this isn t so improbable,. Elementary Statistics: Looking at the Big Picture L19.37
28 Lecture Summary (Distribution of Sample Proportion) Typical inference problem Sampling distribution; definition 3 approaches to understanding sampling dist. Intuition Hands-on experiment Theory Center, spread, shape of sampling distribution Central Limit Theorem Role of sample size Applying Rule Elementary Statistics: Looking at the Big Picture L19.38
of course the mean is p. That is just saying the average sample would have 82% answering
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